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Small Fractional Parts of Polynomials

A co-publication of the AMS and CBMS
Available Formats:
Electronic ISBN: 978-1-4704-2392-6
Product Code: CBMS/32.E
List Price: $26.00 Individual Price:$20.80
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Small Fractional Parts of Polynomials
A co-publication of the AMS and CBMS
Available Formats:
 Electronic ISBN: 978-1-4704-2392-6 Product Code: CBMS/32.E
 List Price: $26.00 Individual Price:$20.80
• Book Details

CBMS Regional Conference Series in Mathematics
Volume: 321977; 41 pp
MSC: Primary 11;

Knowledge about fractional parts of linear polynomials is fairly satisfactory. Knowledge about fractional parts of nonlinear polynomials is not so satisfactory. In these notes the author starts out with Heilbronn's Theorem on quadratic polynomials and branches out in three directions. In Sections 7–12 he deals with arbitrary polynomials with constant term zero. In Sections 13–19 he takes up simultaneous approximation of quadratic polynomials. In Sections 20–21 he discusses special quadratic polynomials in several variables.

There are many open questions: in fact, most of the results obtained in these notes ar almost certainly not best possible. Since the theory is not in its final form including the most general situation, i.e. simultaneous fractional parts of polynomials in several variables of arbitary degree. On the other hand, he has given all proofs in full detail and at a leisurely pace.

For the first half of this work, only the standard notions of an undergraduate number theory course are required. For the second half, some knowledge of the geometry of numbers is helpful.

• Chapters
• Heilbronn’s Theorem
• The Heilbronn Alternative Lemma
• About Sums $\sum \| \xi _i \|^{-1}$
• About Sums $\sum e(\alpha n^2)$
• Proof of the Heilbronn Alternative Lemma
• Fractional Parts of Polynomials
• A General Alternative Lemma
• Sums $\sum \| \xi _i \|^{-1}$ Again
• Estimation of Weyl Sums
• What Happens if the Weyl Sums are Large
• Proof of the General Alternative Theorem
• Simultaneous Approximation
• A Reduction
• Proof of the Alternative Lemma on Simultaneous Approximation
• On max $\| \alpha _in^2 \|$
• A Determinant Argument
• Proof of the Three Alternatives Lemma
• Quadratic Polynomials in Several Variables
• Requests

Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Volume: 321977; 41 pp
MSC: Primary 11;

Knowledge about fractional parts of linear polynomials is fairly satisfactory. Knowledge about fractional parts of nonlinear polynomials is not so satisfactory. In these notes the author starts out with Heilbronn's Theorem on quadratic polynomials and branches out in three directions. In Sections 7–12 he deals with arbitrary polynomials with constant term zero. In Sections 13–19 he takes up simultaneous approximation of quadratic polynomials. In Sections 20–21 he discusses special quadratic polynomials in several variables.

There are many open questions: in fact, most of the results obtained in these notes ar almost certainly not best possible. Since the theory is not in its final form including the most general situation, i.e. simultaneous fractional parts of polynomials in several variables of arbitary degree. On the other hand, he has given all proofs in full detail and at a leisurely pace.

For the first half of this work, only the standard notions of an undergraduate number theory course are required. For the second half, some knowledge of the geometry of numbers is helpful.

• Chapters
• Heilbronn’s Theorem
• The Heilbronn Alternative Lemma
• About Sums $\sum \| \xi _i \|^{-1}$
• About Sums $\sum e(\alpha n^2)$
• Proof of the Heilbronn Alternative Lemma
• Fractional Parts of Polynomials
• A General Alternative Lemma
• Sums $\sum \| \xi _i \|^{-1}$ Again
• Estimation of Weyl Sums
• What Happens if the Weyl Sums are Large
• Proof of the General Alternative Theorem
• Simultaneous Approximation
• A Reduction
• Proof of the Alternative Lemma on Simultaneous Approximation
• On max $\| \alpha _in^2 \|$
• A Determinant Argument
• Proof of the Three Alternatives Lemma
• Quadratic Polynomials in Several Variables